/var/folders/c3/kmslgy_s5633b97ysvr59lnh034hk7/T/ipykernel_98595/1520744162.py:14: DeprecationWarning: `set_matplotlib_formats` is deprecated since IPython 7.23, directly use `matplotlib_inline.backend_inline.set_matplotlib_formats()`
  set_matplotlib_formats('svg')

Introduction to Ambisonics

Fundamentals

Chris Hold, Virtual Acoustics (Aalto University)

Structure

• Motivation
• Discovering Spherical Harmonics
• Ambisonics Encoder and Decoder

From Recording to Playback

Audio Description Formats

Channel Based	Object Based	Scene Based

Benefits of Scene-based

• Separation of recording, transmission / storage, and playback
• Flexible with multiple options for each stage
• Does not scale with the number of sources

Ambisonics

• Implementation of a scene based format
• "Long" history and connection to other sciences
• Use spherical harmonics as basis functions

Example

From the Wave Equation to Spherical Harmonics

Wave equation \begin{equation} \nabla^2 p - \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} = 0 \quad. \end{equation}

In frequency domain, with wave-number $k = \frac{\omega}{c}$ results in the Helmholtz equation \begin{equation} (\nabla^2 + k^2) p = 0 \quad. \end{equation}

Multiple solutions, e.g. a mono chromatic plane wave with amplitude $\hat{A}(\omega)$ in Cartesian coordinates \begin{equation} p(t, x, y, z) = \hat{A}(\omega) e^{i(k_x x + k_y y + k_z z - \omega t)} \quad. \end{equation}

Any solution to the Helmholtz equation can also be expressed in spherical coordinates \begin{equation} p(r, \theta, \phi, k) = \sum_{n = 0}^{\infty} \sum_{m=-n}^{n} \color{darkgreen}{(A_{mn} j_n(kr) + B_{mn} y_n(kr))} \; \color{darkblue}{Y_{n}^{m}(\theta,\phi)} \quad, \end{equation}

\begin{equation} p(r, \theta, \phi, k) = \sum_{n = 0}^{\infty} \sum_{m=-n}^{n} \color{darkgreen}{C_{mn}(kr)} \; \color{darkblue}{Y_{n}^{m}(\theta,\phi)} \quad, \end{equation}

With two separable parts:

• radial component : linear combination of spherical Bessel functions of first ($ j_n $) and second kind $ y_n $
• angular component : spherical harmonics $ \color{darkblue}{Y_{n}^{m}(\theta,\phi)} $

sound field defined on a sphere is fully captured by its spherical harmonics coefficients

E.g. a unit plane wave \begin{equation} p(r, \theta, \phi, k) = \sum_{n = 0}^{\infty} \sum_{m=-n}^{n} 4 \pi i^n j_n(kr) \left[ Y_n^m(\theta_k,\phi_k) \right] ^* Y_{n}^ {m}(\theta,\phi) \quad. \end{equation}

Discovering Spherical Harmonics

\begin{equation} Y_n^m(\theta,\phi)=\color{darkorange}{\sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}}\, \color{darkgreen}{P_n^m(\cos(\theta))}\, \color{darkred}{e^{im\phi}} \quad, \end{equation}\begin{equation} Y_n^m(\theta,\phi)=\color{darkorange}{D_{nm}}\, \color{darkgreen}{P_n^m(\cos(\theta))}\, \color{darkred}{e^{im\phi}} \quad, \end{equation}

where

• azimuth as $\phi$ and zenith/colatitude as $\theta$.
• $P_n^m$ is the associated Legendre polynomial of order $n$ and degree $m$.

Combined with appropriate scaling give real Spherical Harmonics $Y_{n,m}(\theta,\phi)$ as

\begin{equation} Y_{n,m}(\theta,\phi) = \sqrt{ \frac{(2n+1)}{4\pi} \frac{(n-|m|)!}{(n+|m|)!} } P_{n,|m|}(\cos(\theta)) \begin{cases} \sqrt2\sin(|m|\phi) & \mathrm{if\hspace{0.5em}} m < 0 \quad,\\ 1 & \mathrm{if\hspace{0.5em}} m = 0 \quad,\\ \sqrt2\cos(|m|\phi) & \mathrm{if\hspace{0.5em}} m > 0 \quad. \end{cases} \end{equation}

• Always check: Condon–Shortley phase convention $(-1)^m$

Let's look at the azimuthal component $e^{im\phi}$ and the zenithal component $P_n^m(\cos\theta)$

azimuthal component

zenithal component

Orthonormality

Two functions $\color{darkviolet}f, \color{darkred}g$ over a domain $\gamma$ are orthogonal if

\begin{equation} \int_\gamma \color{darkviolet}{f^*(\gamma)} \color{darkred}{g(\gamma)} \,\mathrm{d}\gamma = \langle \color{darkviolet}f, \color{darkred}g \rangle = 0 \quad,\,\mathrm{for}~f \neq g. \end{equation}

They are also orthonormal if \begin{equation} \color{darkviolet}{\int_\gamma f^*(\gamma) f(\gamma) \,\mathrm{d}\gamma = \int_\gamma|f(\gamma)|^2 \,\mathrm{d}\gamma = \langle f, f \rangle } = 1 \quad, \\ \color{darkred}{\int_\gamma g^*(\gamma) g(\gamma) \,\mathrm{d}\gamma = \int_\gamma |g(\gamma)|^2 \,\mathrm{d}\gamma = \langle g, g \rangle} = 1 \quad. \end{equation}

For the Spherical Harmonics:

• the the azimuthal component $e^{im\phi}$ along $\phi$ is orthogonal w.r.t. the degree $m$
• the zenithal component $P_n^m(\cos\theta)$ along $\theta$ is orthogonal w.r.t. the order $n$.

Their product is still orthogonal, and the scaling $\color{darkorange}{D_{nm}}$ ensures orthonormality such that \begin{equation} \int_\Omega Y_n^m(\Omega)^* \, Y_{n'}^{m'}(\Omega) \,\mathrm{d}\Omega = \langle Y_n^m(\Omega) , Y_{n'}^{m'}(\Omega) \rangle = \delta_{nn'}\delta_{mm'} \quad , \end{equation}

and

\begin{equation} \int_{{\Omega} \in \mathbb{S}^2} |Y_n^m({\Omega})|^2 \mathrm{d}{\Omega} = 1 \quad . \end{equation}

Example

show normality for $Y_0^0$: $$ Y_0^0(\theta,\phi)=\sqrt{\frac{0+1}{4\pi}\frac{(0)!}{(0)!}} P_0^0(\cos(\theta)) e^{i0\phi} = \sqrt{\frac{1}{4\pi}} \quad, $$ hence $$ \int_{{\Omega} \in \mathbb{S}^2} Y_0^0(\theta,\phi)^* Y_0^0(\theta,\phi) \,\mathrm{d}{\Omega} = \int_{{\Omega} \in \mathbb{S}^2} \sqrt{\frac{1}{4\pi}}\sqrt{\frac{1}{4\pi}} \, \mathrm{d}{\Omega} = 4\pi \frac{1}{4\pi} = 1 \quad. $$

Example

Test orthogonality of functions $\color{darkviolet}f, \color{darkred}g$

$$ \langle \color{darkviolet}f, \color{darkred}g \rangle \overset{?}{=} 0 \quad,\,\mathrm{for}~f \neq g. $$

Test for orthogonality in azimuth <f_azi, g_azi> =  0.0
Test for orthogonality in zenith <f_zen, g_zen> =  0.0

Spherical Harmonic Transform (SHT)

• Find a scene based encoding for sound field
• We showed that the spherical harmonics $Y_n^m({\Omega})$ are a set of suitable basis functions on the sphere
• We also showed that a sound field (on the sphere) $s({\Omega})$ is fully captured by its spherical harmonics coefficients $\sigma_{nm}$

This can be expressed with $\Omega = [\phi, \theta]$ as the inverse Spherical Harmonic Transform (iSHT) \begin{equation} s({\Omega}) = \sum_{n = 0}^{N=\infty} \sum_{m=-n}^{+n} \sigma_{nm} Y_n^m({\Omega}) \quad. \end{equation}

Spherical harmonics coefficients $\sigma_{nm}$ can be derived with the Spherical Harmonic Transform (SHT) \begin{equation} \sigma_{nm} = \int_{{\Omega} \in \mathbb{S}^2} s({\Omega}) [Y_n^m({\Omega})]^* \mathrm{d}{\Omega} = \langle [Y_n^m({\Omega})] , s({\Omega}) \rangle \quad. \end{equation}

• SHT also referred to as spherical Fourier transform
• Note the transform from a spatially discrete to continuous domain

Spherical Grids

The SHT evaluates the continuous integral over $\Omega$ \begin{equation} \sigma_{nm} = \int_{{\Omega} \in \mathbb{S}^2} s({\Omega}) [Y_n^m({\Omega})]^* \mathrm{d}{\Omega} \quad . \end{equation} Quadrature methods allow evaluation by spherical sampling at certain (weighted) grid points such that \begin{equation} \sigma_{nm} \approx \sum_{q=1}^{Q} w_q s({\Omega}_q) [Y_n^m({\Omega}_q)]^* \quad. \end{equation}

Certain grids with sampling points $ {\Omega}_q $ and associated sampling weights $w_q$ have certain properties:

• quadrature grids allow numerical integration of spatial polynomials
• Spatial sampling dictates maximum order
• easy to evaluate are uniform/regular grids, with constant $w_q = \frac{4\pi}{Q}$
• An example are so-called spherical t-designs(t), which allow a SHT up to order $N = \lfloor t/2 \rfloor$

Spatial Dirac

We can show that spherical harmonics are orthogonal (even orthonormal) with \begin{equation} \int_\Omega Y_n^m(\Omega) \, Y_{n'}^{m'}(\Omega) \,\mathrm{d}\Omega = \delta_{nn'}\delta_{mm'} \quad . \end{equation}

Because of their completeness, we can also directly formulate a spatial Dirac function on the sphere as \begin{equation} \sum_{n=0}^{N=\infty} \sum_{m=-n}^n [Y_n^m({\Omega'})]^* Y_n^m(\Omega) = \delta(\Omega - \Omega') \quad, \end{equation}

and therefore the spherical Fourier coefficients $\sigma_{nm}$ \begin{equation} SHT\{\delta(\Omega - \Omega')\} = \int_{{\Omega} \in \mathbb{S}^2} \delta(\Omega - \Omega') \, [Y_n^m({\Omega})]^* \mathrm{d}{\Omega} = [Y_n^m({\Omega'})]^* \quad . \end{equation}

Order-Limitation of Spatial Dirac Pulse

Example

Integrate (order-limited) Spatial Dirac $\delta_N(\Omega - \Omega')$ over sphere

$$ \int_{{\Omega} \in \mathbb{S}^2} \delta_N(\Omega - \Omega') \mathrm{d}{\Omega} = \int_{{\Omega} \in \mathbb{S}^2} \color{darkblue}{\sum_{n=0}^{N} \sum_{m=-n}^n\, [Y_n^m({\Omega'})]^* Y_n^m(\Omega)} \,\mathrm{d}{\Omega} \quad, $$

by discretization with sufficient t-design $$ \int_{{\Omega} \in \mathbb{S}^2} \color{darkblue}{\sum_{n=0}^{N} \sum_{m=-n}^n\, [Y_n^m({\Omega'})]^* Y_n^m(\Omega)} \,\mathrm{d}{\Omega} = \frac{4\pi}{Q}\sum_{q=1}^{Q} \color{darkorange}{\sum_{n=0}^{N} \sum_{m=-n}^n\, [Y_n^m({\Omega'})]^* Y_n^m(\Omega_q)} $$

Matrix Notations

Stack the spherical harmonics evaluated at $\Omega$ up to spherical order $N$ as

$$ \mathbf{Y} = \left[ \begin{array}{ccccc} Y_0^0(\Omega[0]) & Y_1^{-1}(\Omega[0]) & Y_1^0(\Omega[0]) & \dots & Y_N^N(\Omega[0]) \\ Y_0^0(\Omega[1]) & Y_1^{-1}(\Omega[1]) & Y_1^0(\Omega[1]) & \dots & Y_N^N(\Omega[1]) \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ Y_0^0(\Omega[Q-1]) & Y_1^{-1}(\Omega[Q-1]) & Y_1^0(\Omega[Q-1]) & \dots & Y_N^N(\Omega[Q-1]) \end{array} \right] $$

such that $ \mathbf{Y} : [Q, (N+1)^2]$

• soundfield pressure is real signal -> only real spherical harmonics needed
• orthonormal scaling introduced earlier called N3D by convention (in contrast to SN3D)
• stacking them as above, where $idx_{n,m} = n^2+n+m$, is called ACN

Obtaining the discrete signals $s_q(t)$ is a linear combination of SH basis functions evaluated at $\Omega_q$.

This inverse transform in matrix notation with ambisonic signals matrix $\mathbf{\chi} : [1, (N+1)^2]$ is

\begin{equation} s_q(t) = \mathbf{\chi}(t) \, \mathbf{Y}^T \quad . \end{equation}

We obtain ambisonic signals matrix $\chi : [1, (N+1)^2]$ from signals $\mathbf{S} : [1, Q]$ by SHT as

\begin{equation} \mathbf{\chi}(t) = \mathbf{S}(t) \, \mathrm{diag}(w_q) \, \mathbf{Y} \quad. \end{equation}

From the Ambisonics Encoder, to Spatial Weighting, to a Decoder

Encoder

• We can stack a time signal $s(t_0, t_1, \ldots, t_n)$ as vector $\mathbf{s}$.
• This leads to a matrix notation of multiple discrete signals $\mathbf{S} : [t, Q]$.
• Similarly, we can stack the Ambisonics coefficients into matrix $\mathbf{\chi} : [t, (N+1)^2]$

A single plane wave encoded in direction $\Omega$ with signal $\mathbf{s}$ is directly the outer product with the spatial Dirac coefficients

\begin{equation} \mathbf{\chi}_{PW(\Omega)} = \mathbf{s} \, \mathbf{Y}(\Omega) \quad. \end{equation}

For multiple sources $Q$, we stack and sum \begin{equation} \mathbf{\chi}_{PW(\Omega_Q)} = \sum_{q=1}^Q\mathbf{s}_q \, \mathbf{Y}(\Omega_q) = \mathbf{S} \, \mathbf{Y} \quad. \end{equation}

Spatial Weighting

• Directionally filtering a soundfield in direction $\Omega_k$ by spatial weighting
• Weighting in SH domain $w_{nm}$ is an elegant way of beamforming

The simplest beamformer is a spatial Dirac in direction $\Omega_k$ normalized by its energy, i.e. $\mathrm{max}DI$ \begin{equation} w_{nm, \mathrm{max}DI}(\Omega_k) = \frac{4\pi}{(N+1)^2} Y_{n,m}(\Omega_k) \end{equation}

Other patterns can be achieved by weighting the spherical Fourier spectrum. Axis-symmetric patterns reduce to only a modal weighting $c_n$, such that \begin{equation} w_{nm}(\Omega_k) = c_{n} \, Y_{n,m}(\Omega_k) \end{equation}

E.g. $\mathrm{max}\vec{r}_E$ weights each order with \begin{equation} c_{n,\,\mathrm{max}\vec{r}_E} = P_n[\cos(\frac{137.9^\circ}{N+1.51})] \quad, \end{equation} with the Legendre polynomials $P_n$ of order $n$.

Or we can define a (spatial) Butterworth filter with \begin{equation} c_{n,\,\mathrm{Butterworth}} = \frac{1}{\sqrt{1+(n/n_c)^{2k}}} \quad, \end{equation} with the filter order $k$ and cut-on $n_c$.

Decoder

• Pointing a set of beamformers in directions $\Omega_k$ results in a set of signal components
• in case of $\mathrm{max}DI$ proportional to iSHT in $\Omega_k$

\begin{equation} s({\Omega_k}) = \sum_{n = 0}^{N} \sum_{m=-n}^{+n} w_{nm}({\Omega_k}) \, \sigma_{nm} \quad, \end{equation}

or in matrix notation with $\mathbf{S} : [t, Q]$ and beamforming weights $\mathbf{c}_n$ \begin{equation} \mathbf{S} = \mathbf{\chi} \, \mathrm{diag_N}(\mathbf{c}_n) \, \mathbf{Y}^T \quad . \end{equation}

Example

Decode on a t-design(6) (sufficient up to $N = 3$):

• a 3rd order ambisonic signal

Example

Decode on a t-design(6) (sufficient up to $N = 3$):

• a 5th order ambisonic signal
• an 8th order ambisonic signal

Loudspeaker Decoders

# Loudspeaker Setup
ls_dirs = np.array([[-135, -80, -45, 0, 45, 80, 135, -135, -60, -30, 30, 60, 135],
                    [0, 0, 0, 0, 0, 0, 0, 60, 60, 60, 60, 60, 60]])
ls_x, ls_y, ls_z = spa.utils.sph2cart(spa.utils.deg2rad(ls_dirs[0, :]),
                                      spa.utils.deg2rad(90 - ls_dirs[1, :]))
ls_setup = spa.decoder.LoudspeakerSetup(ls_x, ls_y, ls_z)
ls_setup.pop_triangles(normal_limit=85, aperture_limit=90,
                       opening_limit=150)
ls_setup.show()

Face not pointing towards listener: [0 1 2]
Face not pointing towards listener: [0 6 2]
Face not pointing towards listener: [2 5 6]
Face not pointing towards listener: [2 3 4]
Face not pointing towards listener: [2 5 4]

spa.plots.decoder_performance(ls_setup, 'NLS')

spa.plots.decoder_performance(ls_setup, 'VBAP')

spa.plots.decoder_performance(ls_setup, 'SAD')

spa.plots.decoder_performance(ls_setup, 'MAD')

spa.plots.decoder_performance(ls_setup, 'EPAD', N_sph=2)

ls_setup.ambisonics_setup(update_hull=True)
spa.plots.decoder_performance(ls_setup, 'ALLRAD')

ls_setup.ambisonics_setup(update_hull=True)
spa.plots.decoder_performance(ls_setup, 'ALLRAD2')

Binaural Decoding

\begin{equation} s^{l,r}(t) = x(t) * h_{\mathrm{HRIR}}^{l,r}({\Omega}, t) \quad, \end{equation}

where $(*)$ denotes the time-domain convolution operation.

Transforming to the time-frequency domain through the time-domain Fourier transform, further assuming plane-wave components $\bar X (\Omega)$, the ear input signals are given as \begin{equation} S^{l,r}(\omega) = \int_{\Omega} \bar X (\Omega, \omega) H^{l,r}(\Omega, \omega) \,\mathrm{d}\Omega \quad. \end{equation}

\begin{equation} S^{l,r}(\omega) = \sum_{n = 0}^{N} \sum_{m = -n}^{+n} \chi_{nm}(\omega) \breve H_{nm}^{l,r}(\omega) \quad. \end{equation}

For one ear (left) this can be interpreted as a frequency dependent ambisonic beamformer \begin{equation} s^l(\omega) = \chi_{nm}(\omega) [\breve H_{nm}^{l}(\omega)]^T \quad . \end{equation}